Question

let a and b be subsets of a universal set u and suppose n(u) = 300, n(a) = 150, n(b) = 120, and n(a ∩ b) = 60. compute: (a) n(a^c ∩ b) (b) n(b^c) (c) n(a^c ∩ b^c)

Explanation:

Step1: Calculate $n(A^c \cap B)$

Use the formula $n(A^c \cap B) = n(B) - n(A \cap B)$
$n(A^c \cap B) = 120 - 60 = 60$

Step2: Calculate $n(B^c)$

Use the formula $n(B^c) = n(U) - n(B)$
$n(B^c) = 300 - 120 = 180$

Step3: Calculate $n(A^c \cap B^c)$

First find $n(A \cup B)$ with $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(A \cup B) = 150 + 120 - 60 = 210$
Then use $n(A^c \cap B^c) = n(U) - n(A \cup B)$
$n(A^c \cap B^c) = 300 - 210 = 90$

Answer:

(a) $60$
(b) $180$
(c) $90$